Polynomial Maximal Surfaces in H^{2,2} and Spherical Polygons
Abstract
We study the moduli space of complete maximal surfaces in pseudo-hyperbolic space H2,2 with polygonal asymptotic boundary a lightlike (n+4)-gon in the Einstein universe Ein1,2, up to the action of SO0(2,3). Importantly, this space Mn+4(H2,2) is disconnected for n≥ 2. We introduce a \emph{turning invariant} Θ that classifies the components in terms of the geometry of the boundary polygons in the Einstein universe Ein1,2, which is related to \emph{spherical polygons} in 𝕊2. We provide a geometric parametrization of the moduli space Mn+4(H2,2) via holomorphic data, namely a pair of polynomials α, β. These polynomials appear in relation to certain components of the complexified second fundamental form of the corresponding surface. We then relate the holomorphic data α,β with the turning invariant Θ. This leads to characterizations of the \emph{Hitchin component} of polygonal maximal surfaces studied by Tamburelli-Wolf.
Related
- Tilings of Closed Surfaces by Congruent Most Curvilinear Polygons
- Bilinear Korányi spherical maximal function on the Heisenberg group
- Endpoint power weight inequalities for spherical maximal functions
- Sharp Lp estimates for the strong spherical maximal operator
- Spherical harmonics with maximal Lp norm growth
BibTeX
@misc{evans2026polynomial,
title = {{Polynomial Maximal Surfaces in H^2,2 and Spherical Polygons}},
author = {Parker Evans and Alex Moriani},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.22329},
eprint = {2610.22329},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.22329},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Polynomial Maximal Surfaces in H^{2,2} and Spherical Polygons
AU - Parker Evans
AU - Alex Moriani
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.22329
UR - https://arxiv.org/abs/2610.22329
AB - We study the moduli space of complete maximal surfaces in pseudo-hyperbolic space H^2,2 with polygonal asymptotic boundary a lightlike (n+4)-gon in the Einstein universe Ein^1,2, up to the action of SO_0(2,3). Importantly, this space M_n+4(H^2,2) is disconnected for n≥ 2. We introduce a \emph{turning invariant} Θ that classifies the components in terms of the geometry of the boundary polygons in the Einstein universe Ein^1,2, which is related to \emph{spherical polygons} in 𝕊^2. We provide a geometric parametrization of the moduli space M_n+4(H^2,2) via holomorphic data, namely a pair of polynomials α, β. These polynomials appear in relation to certain components of the complexified second fundamental form of the corresponding surface. We then relate the holomorphic data α,β with the turning invariant Θ. This leads to characterizations of the \emph{Hitchin component} of polygonal maximal surfaces studied by Tamburelli-Wolf.
ER -